Delta Hedging, Volatility Skew, and Skew Stickiness Ratio

Summary

The document examines how a derivative’s hedge changes when implied volatility depends on the underlying price. It states that the total sensitivity to the underlying combines the usual partial delta with a vega contribution that depends on how volatility moves as the underlying changes. It then asks whether the skew stickiness ratio, defined using the change in forward volatility against log spot and the volatility skew, can be used to infer that volatility sensitivity.

The author identifies two apparent complications: the logarithmic spot transformation affects the derivative relationship, and sticky-strike behavior is described as having zero volatility change with spot even though the ratio is one. The text does not resolve these points or supply a derivation, model, or empirical test. It is useful as a conceptual prompt about translating volatility-surface dynamics into hedge adjustments, but should not be treated as an established SSR-to-delta formula.

Key ideas

• When volatility depends on the underlying, total delta includes a vega-weighted volatility sensitivity.
• The skew stickiness ratio relates forward volatility changes to log spot moves and the volatility skew.
• The document questions how the logarithmic spot transformation affects the inferred volatility sensitivity.
• It notes an apparent tension between sticky-strike behavior and the stated SSR value.
• No derivation or resolution is provided.

Tags

# How does delta adjustment relate to skew stickiness ratio (SSR)?


# How does delta adjustment relate to skew stickiness ratio (SSR)?












The correct delta hedging of a derivative $V$ in a model where volatility $\sigma$ is a function of the underlier $S$ requires a stock holding of an amount $$ \frac{dV}{dS}=\frac{\partial V}{\partial S}+\frac{\partial V}{\partial \sigma}\frac{\partial \sigma}{\partial S}. $$ If I understand correctly, this article makes the point that by thinking about volatility dynamics in terms of a volatility level and a shape curve, it is possible to deduce the $\partial \sigma/\partial S$ from the Skew Stickiness Ratio (SSR) defined by Bergomi (2004) and Bergomi (2016) as $$ \mathcal{R}=\frac{1}{\mathcal{S}}\frac{d\sigma_F}{d\log S}, $$ where $\sigma_F$ is the at the forward volatility for a certain maturity and $\mathcal{S}=\partial\sigma_F/\partial K$ is the volatility skew for the same maturity. Can anybody explain me how would it be possible? $\partial\sigma/\partial S=\mathcal{S R}$ does not hold due to the logarithmic transformation of $S$. In addition, we know from different perspectives that under the sticky-strike assumption $d\sigma/d S=0$ while $\mathcal{R}=1$. Can anybody help me make sense of this?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.